Year: 2026
Type: preprint
Abstract: Assume that the Riemann zeta function has no zeros in $\Re(s)>1-δ$, $s\neq1$, for some fixed $0<δ\leq1/2$. We prove that, for every sufficiently large integer $k$, a positive multiple $m_k k!$ has arithmetic complexity $τ(m_k k!)\leq\exp(C\sqrt{\log k\log\log k})$, where $C$ depends only on $δ$. We first establish a smooth-number asymptotic at every sufficiently large center $x$ in intervals of length at least $x^{3/4}$. Combining this estimate with an established genus-two construction yields the arithmetic bound without Cheng's conjecture on smooth numbers in elliptic Hasse intervals. The bound includes the cost of constructing all integer parameters. This work was carried out with assistance from the UniMath model, developed by UniDT (Shanghai) Co., Ltd., and OpenAI's GPT-6.1 Sol and GPT-6 Astra.
Source: arXiv (Cornell University)
Author: Zhang, Liang
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7224128606